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A theory-based decision heuristic for DPLL(T)

Dan Goldwasser Ofer Strichman Shai Fine Haifa university TechnionIBM-HRL

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DPLL

Decide

BCPAnalyze conflict

Backtrack

SAT

UNSAT

full assignment

partial assignment

conflict

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DPLL(T)

Decide

BCP

Deduction Add Clauses

Analyze conflict

Backtrack

SAT

UNSAT

full assignment

partial assignment

conflict

T-propagation / T-conflict

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Theory propagation

Matters for efficiency, not correctness. Depending on the theory, the best strategy can

be: One T-implication at a time All possible T-implications (“exhaustive theory-

propagation”). Cheap-to-compute T-implications

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In this work we are interested in Linear Arithmetic (LA)

We will see: The potential of theory propagation Why doesn’t it work today How can it be approximated efficiently

Theory propagation for LA

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A geometric interpretation

Let H be a finite set of hyperplanes in d dimensions. Let n = |H|

An arrangement of H, denoted A(H), is a partition of Rd.

An arrangement in d=2:

# cells · nd

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A geometric interpretation

Consider a consistent partial assignment of size r. e.g. assignment to (l1,l2,l3), hence r =3.

How many such T-implications are there ?

2l

3l

l1l4

current partial assignment

(1,0,0)

n = 6r = 3

l5

T-Implied

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A geometric interpretation

Consider a consistent partial assignment of size r .

Theorem 1: O((n ¢ log r) /r) of the remaining constraints intersect the cell [HW87] with high probability (1 - 1/rc).

Some example numbers: r = 3, ~47% of the remaining constraints are implied. r = 12, ~70% of the remaining constraints are implied. r = 60, ~90% of the remaining constraints are implied.

[HW87] D. Haussler and E. Welzl. Epsilon-nets and simplex range queries. Comput. Geom., 2:127- 151, 1987.

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Assigned vs. implied in practice

Two benchmarks. Measured averages at T-consistent points

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Theory propagation for LA

Let l1, l2, l3 be asserted. Is l4 (or :l4) T-implied ?

Two techniques for finding T-implications.

1.“Plunging”: check satisfiability of (l1 Æ l2 Æ l3 Æ l4) and of (l1 Æ l2 Æ l3 Æ :l4)

Requires solving a linear system.

Too expensive in practice (see e.g. [DdM06]).

[DdM06] Integrating simplex with DPLL(T), Dutertre and De Moura, SRI-CSL-06-01

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Theory propagation for LA

Let l1, l2, l3 be asserted. Is l4 (or :l4) T-implied ?

Two techniques for finding T-implications.

2. Check if all vertices on the same side of l4

There is an exponential number of vertices.

Too expensive in practice.

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Approximating theory propagation

Problem 1: How can we use conjectured information without losing soundness ?

Problem 2: how can we find cheaply good conjectures i.e., conjectured T-implications

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Problem 1: how to use conjectures ? We use conjectured implications just to bias

decisions. SAT chooses a variable to decide, we conjecture

its value.

Might be better than the alternative: SAT’s heuristics are T-ignorant.

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Problem 2: conjecturing T-implications We examined two methods: 1. k - vertices

Find k-vertices. If they are all on the same side of l4 – conjecture that l4

is implied.

l4

In this case we conjecture :l4

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Problem 2: conjecturing T-implications We examined two methods: 1. k - vertices

Find k-vertices. If they are all on the same side of l4 – conjecture that l4

is implied.

l4

In this case we conjecture nothing

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Problem 2: conjecturing T-implications We examined two methods: 1. k - vertices

Find k-vertices. If they are all on the same side of l4 – conjecture that l4

is implied.

l4

In this case we (falsely) conjecture l4

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Problem 2: conjecturing T-implications We examined two methods: 1. k - vertices

Find k-vertices. If they are all on the same side of l4 – conjecture that l4

is implied.

Too expensive in practice

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Problem 2: conjecturing T-implications We examined two methods: 2.One approximated point

Here we always conjecture a T-implication.

l4

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Problem 2: conjecturing T-implications We examined two methods: 2.One approximated point

Here we always conjecture a T-implication.

l4

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Problem 2: conjecturing T-implications We examined two methods: 2.One approximated point

Here we always conjecture a T-implication.

l4

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Problem 2: conjecturing T-implications We examined two methods: 2.One approximated point

The idea: use the assignment maintained by Simplex. It’s for free.

Competitive SMT solvers Use general Simplex [DdM06], not classical Simplex Do not activate Simplex after each assignment They only update the assignment according to the

‘simple’ constraints (e.g. “x < c”).

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Problem 2: conjecturing T-implications The assignment maintained by general Simplex is

updated after each partial (Boolean) assignment Based on simple constraints only.

Several possibilities:

is T-inconsistent

is T-consistent doesn’t satisfy it

is T-consistent satisfies it

22%

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Problem 2: conjecturing T-implications Our hope: is ‘close’ to the polygon. Therefore it can be successful in guessing

implications. Even if l4 is not T-implied, it can guide the search.

l4

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Results

Some results for the 200 benchmarks from SMT-COMP’07

Implementation on top of ArgoLib

Each column refers to a different strategy of choosing the value.

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0-pivot vs. MinisatM

iniS

at

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Back to the future # of cells is exponential in d rather than exponential in

n nd rather than 2n

for n sufficiently larger than d, better worst-case complexity

SMT-LIB + SRI’s GDP benchmarks Examples: n : d

QF_RDL_SCHEDULING 10.9 : 1 QF_RDL_SAL 6.7 : 1 QF_LRA_SC 3.9 : 1 QF_LRA_START_UP 6.9 : 1 QF_LRA_UART 6.1 : 1 QF_LRA_CLOCK_SYNCH 3.3 : 1 QF_LRA_SPIDER_BENCHMARKS 3.2 : 1 QF_LRA_SAL 6.1 : 1 MathSAT benchmarks (difference logic) 44.5 : 1 SEP benchmarks (difference logic) 17 : 1

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P#2: a reversed lazy approach

Current SAT-based ‘lazy’ approaches Search the Boolean domain check assignment in the

theory domain A ‘reversed lazy approach’:

Search the theory domain check assignment in the Boolean domain

T-solver

SAT

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How can we enumerate the cells ?

There exists a data structure (“incidence graph”) that represents the linear arrangement

Too large in practice… Corresponds to an explicit representation of the search

space. Constructing a symbolic representation seems as hard as

building the arrangement.

For two years we worked on a random, incremental algorithm, each time adding a constraint and consulting SAT.

The short summary: we were unable to beat Yices…

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Summary

We showed how to use ‘free’ information computed by general simplex in order to improve SAT’s decision. Somewhat compensates on the fact that there is no

theory propagation.

Future research: How can we let the theory lead efficiently ?